Rigid Blade (Flap and Lag)

R2026b

Simulate flap and lag motion of rotor blades

Since R2026b

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Rigid Blade (Flap and Lag) block

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Libraries:
Aerospace Blockset / Rotor Systems / Blade Components

Description

The Rigid Blade (Flap and Lag) block models coupled flap and lag dynamics or isolated flap and lag dynamics of rigid rotor blades in response to distributed aerodynamic forces. The outputs of this block can feed Blade Element Theory and other aerodynamic blocks for rotorcraft analysis.

Limitations

The block does not support simulation in Accelerator mode, Rapid Accelerator mode, or code generation.

Ports

Input

f_z — Normal force distribution

real array

Normal force distribution, specified as a real array of size N_b-by-N_r, where N_b is the number of blades and N_r is the number of radial locations along the blade span. When forces are computed using the Blade Element Theory block, it is expected that the force values provided correspond to mid points of the radial segments used in the computation.

Dependencies

To enable this port, set the Model parameter to Coupled flap and lag or Isolated flap.

Data Types: double

f_x — Tangential force distribution

real array

Tangential force distribution, specified as a real array of size N_b-by-N_r, where N_b is the number of blades and N_r is the number of radial locations along the blade span. When forces are computed using Blade Element Theory, it is expected that the force values provided correspond to mid points of the radial segments used in the computation.

Dependencies

To enable this port, set the Model parameter to Coupled flap and lag or Isolated lag.

Data Types: double

Ω — Angular velocity of rotor

real scalar

Angular velocity of the rotor, specified as a real scalar in radians per second.

Dependencies

To enable this port, set the Rotor rotational speed source parameter to Port.

Data Types: double

Output

β — Blade flap angle

N_b×1 vector

Blade flap angle, returned as a real vector of size N_b in radians. It represents the out‑of‑plane angular deflection of each blade from the hub plane. Flap-up motion is considered positive.

Dependencies

To enable this port, set the Model parameter to Coupled flap and lag or Isolated flap.

Data Types: double

dβ/dt — Blade flap rate

N_b×1 vector

Blade flap rate, returned as a real vector of size N_b in radians per second. It represents the out‑of‑plane angular velocity of each blade.

Dependencies

To enable this port, set the Model parameter to Coupled flap and lag or Isolated flap.

Data Types: double

ζ — Blade lag angle

N_b×1 vector

Blade lag angle, returned as a real vector of size N_b in radians. It represents the in‑plane angular displacement (lead–lag motion) of each blade relative to the hub. Lag motion is considered positive.

Dependencies

To enable this port, set the Model parameter to Coupled flap and lag or Isolated lag.

Data Types: double

dζ/dt — Blade lag rate

N_b×1 vector

Blade lag rate, returned as a real vector of size N_b in radians per second. It represents the in‑plane angular velocity of each blade.

Dependencies

To enable this port, set the Model parameter to Coupled flap and lag or Isolated lag.

Data Types: double

Parameters

To edit block parameters interactively, use the Property Inspector. From the Simulink^® Toolstrip, on the Simulation tab, in the Prepare gallery, select Property Inspector.

Main

Units — Input and output units

Metric (MKS) (default) | English

Input and output ports and parameter units, specified as Metric (MKS) or English.

Units	Radius	Elemental Force Distribution	Flap/Lag Inertia	Flap Spring Constant	Flap/Lag Damping
Metric (MKS)	Meters	N/m	kg·m2	N·m/rad	N·m·s/rad
English	Feet	lbf/ft	slug·ft2	lbf·ft/rad	lbf·ft·s/rad

Programmatic Use

Block Parameter: units

Type: character vector

Values: Metric (MKS) | English

Default: Metric (MKS)

Model — Blade dynamics model to simulate

Coupled flap and lag (default) | Isolated flap | Isolated lag

Blade dynamics model to simulate, specified as one of

• Coupled flap and lag — Solves the fully coupled flap and lag dynamics and enables all input and output ports.

• Isolated flap — Solves only the out‑of‑plane flap dynamics. Enables the fz input and the β and dβ/dt outputs.

• Isolated lag — Solves only the in‑plane lag dynamics. Enables the fx input and the ζ and dζ/dt outputs.

Programmatic Use

Block Parameter: model

Type: character vector

Values: Coupled flap and lag | Isolated flap | Isolated lag

Default: Coupled flap and lag

Rotor rotational speed source — Source of angular velocity of the rotor

Dialog (default) | Port

Source of the angular velocity of the rotor, specified as one of Dialog or Port.

Programmatic Use

Block Parameter: omegaSrc

Type: character vector

Values: Dialog | Port

Default: Dialog

Rotor rotational speed — Rotational speed of rotor

41.2596 (default) | real scalar

Rotational speed of rotor, specified as a real scalar in radians per second.

Dependencies

This parameter is visible when the Rotational Speed Source parameter is set to Dialog.

Programmatic Use

Block Parameter: omega

Type: double

Values: 41.2596 | real scalar

Default: 41.2596

Number of blades — Number of rotor blades

4 (default) | positive integer scalar

Number of rotor blades, specified as a positive integer scalar. This parameter defines the blade count used to size the input force arrays and the output angle and rate vectors.

Programmatic Use

Block Parameter: Nb

Type: double

Values: positive integer scalar

Default: 4

Data Types: double

Rotor radius — Rotor radius

5.08 m (default) | real positive scalar

Rotor radius, specified as a nonzero scalar in m or ft. This is the distance from the center of the rotor hub to the tip of the rotor blades.

Dependencies

The unit of Blade radius depends on the value of the Units parameter.

Programmatic Use

Block Parameter: radius

Type: double

Values: 5.08 m | real positive scalar

Default: 5.08 m

Radial locations source — Source of radial locations

Uniform distribution based on force input (default) | Custom distribution

Source of radial locations, specified as one of Uniform distribution based on force input or Custom distribution. This parameter determines how the radial positions along the rotor blade are specified for aerodynamic computations.

• Uniform distribution based on force input - The entire length of the rotor blade is divided into equal segments based on the number of radial locations N_r specified in the force input. For example, if the number of columns in force input is 10, the blade span will be divided into 10 equal segments, as shown in the figure.

  

• Custom distribution - You can specify non-uniform radial positions along the blade, using the parameter Non-dimensional radial element edges (default: -1 for uniform distribution).

Programmatic Use

Block Parameter: radSrc

Type: char vector

Values: Uniform distribution based on force input | Custom distribution

Default: Uniform distribution based on force input

Non-dimensional radial element edges (default: -1 for uniform distribution) — Custom radial locations

-1 (default) | real scalar | real nonnegative strictly increasing vector with values in [0 1]

Custom radial segment limits (non-dimensional), specified as the sentinel value -1 (default) to use uniform distribution, or as a real nonnegative strictly increasing vector with values in [0 1] for custom distribution. The length of the vector should be one greater than N_r, the number of columns in the force input. It is expected that the input force values are computed or obtained at the mid-points of the radial segments specified through the parameter. For example, if the user inputs the parameter as, [r₀ r₁ r₂ r₃ r₄ r₅ r₆ r₇ r₈ r₉ r₁₀], with r₀>=0, and r₁₀<=1, a sample blade segment distribution is as shown in the figure and N_r will be 10. It is expected that the range r₀ to r₁₀ is the aerodynamically effective region of the blade and the integration to compute force is done only in this region. The accuracy of the computed inflow distribution can be influenced by the radial discretization, including the number and spacing of blade elements, as well as the portion of the aerodynamically effective blade span represented by the specified radial segments.

Dependencies

This parameter is visible when Radial locations source is set to Custom distribution.

Programmatic Use

Block Parameter: radDistr

Type: double

Values: real scalar | real nonnegative strictly increasing vector with values in [0 1]

Default: -1

Blade

Flap hinge offset (non-dimensional) — Flap hinge offset

0.06 (default) | real scalar

Flap hinge offset, specified as a real scalar in the range 0 to 1. This represents the distance from the hub center to the physical or effective flap hinge, expressed as a fraction of blade radius. This offset affects the centrifugal stiffening and flapping dynamics of the blade.

Dependencies

This parameter is visible when the Model parameter is set to Coupled flap and lag or Isolated flap.

Programmatic Use

Block Parameter: flapHingeOffset

Type: double

Values: real scalar

Default: 0.06

Data Types: double

Flap inertia — Flap hinge inertia

1084.7 (default) | real positive scalar or vector of length N_b

Flap hinge inertia, specified as a real positive scalar or vector of length N_b. This represents the mass moment of inertia of the blade about the flap hinge, and is its resistance to out‑of‑plane angular acceleration. The unit of this parameter depends on the Units parameter.

Dependencies

This parameter is visible when the Model parameter is set to Coupled flap and lag or Isolated flap.

Programmatic Use

Block Parameter: flapInertia

Type: double

Values: real positive scalar or vector of length Nb

Default: 1084.7

Data Types: double

Flap frequency — Nondimensional flap natural frequency

1.04 (default) | real positive scalar or vector of length N_b

Nondimensional flap natural frequency of out‑of‑plane (flapping) motion, in per rev, specified as a real positive scalar or vector of length N_b. This value is based on structural properties and excludes aerodynamic excitation.

Dependencies

This parameter is visible when the Model parameter is set to Coupled flap and lag or Isolated flap.

Programmatic Use

Block Parameter: flapFrequency

Type: double

Values: real positive scalar or vector of length Nb

Default: 1.04

Flap spring constant — Flap hinge stiffness

0 (default) | real nonnegative scalar or vector of length N_b

Flap spring constant, specified as a real nonnegative scalar or vector of length N_b. This represents the restoring torque per unit flap angle at the flap hinge. The unit of this parameter depends on the Units parameter.

Dependencies

This parameter is visible when the Model parameter is set to Coupled flap and lag or Isolated flap.

Programmatic Use

Block Parameter: flapStiffness

Type: double

Values: real nonnegative scalar or vector of length Nb

Default: 0

Flap damping — Flap hinge damping coefficient

0 (default) | real nonnegative scalar or vector of length N_b

Flap damping coefficient, specified as a real nonnegative scalar or vector of length N_b. This represents the viscous damping coefficient at the flap hinge, producing a retarding torque proportional to flap angular velocity. Aerodynamic damping is not included. The unit of this parameter depends on the Units parameter.

Dependencies

This parameter is visible when the Model parameter is set to Coupled flap and lag or Isolated flap.

Programmatic Use

Block Parameter: flapDamping

Type: double

Values: real nonnegative scalar or vector of length Nb

Default: 0

Precone angle — Precone angle

2.5 (default) | real nonnegative scalar

Precone angle, specified as a real nonnegative scalar in degrees. This represents the fixed upward angle of the blade relative to the hub plane. This parameter is effective only when the flap spring constant is nonzero.

Dependencies

This parameter is visible when the Model parameter is set to Coupled flap and lag or Isolated flap.

Programmatic Use

Block Parameter: precone

Type: double

Values: real nonnegative scalar

Default: 2.5

Lag hinge offset (non‑dimensional) — Lag hinge offset

0.06 (default) | real scalar

Lag hinge offset, specified as a real scalar in the range 0 to 1. This represents the offset from hub center to the physical or effective lag hinge, expressed as a fraction of blade radius. This offset affects the centrifugal stiffening and lag dynamics of the blade.

Dependencies

This parameter is visible when the Model parameter is set to Coupled flap and lag or Isolated lag.

Programmatic Use

Block Parameter: lagHingeOffset

Type: double

Values: real scalar

Default: 0.06

Lag inertia — Lag hinge inertia

1084.7 (default) | real positive scalar or vector of length N_b

Lag hinge inertia, specified as a real positive scalar or vector of length N_b. This represents the mass moment of inertia about the lag hinge, and is its resistance to angular acceleration in the in‑plane (lead‑lag) direction. The unit of this parameter depends on the Units parameter.

Dependencies

This parameter is visible when the Model parameter is set to Coupled flap and lag or Isolated lag.

Programmatic Use

Block Parameter: lagInertia

Type: double

Values: real positive scalar or vector of length Nb

Default: 1084.7

Data Types: double

Lag frequency — Nondimensional lag natural frequency

0.29 (default) | real positive scalar or vector of length N_b

Nondimensional lag natural frequency of in‑plane (lead–lag) motion, in per rev, specified as a real positive scalar or vector of length N_b. This value is based on structural properties and excludes aerodynamic excitation.

Dependencies

This parameter is visible when the Model parameter is set to Coupled flap and lag or Isolated lag.

Programmatic Use

Block Parameter: lagFrequency

Type: double

Values: real positive scalar or vector of length Nb

Default: 0.29

Lag damping — Lag hinge damping coefficient

4067.5 (default) | real nonnegative scalar or vector of length N_b

Damping coefficient at the lag hinge, specified as a real nonnegative scalar or vector of length N_b, providing mechanical damping torque proportional to lag angular velocity. Aerodynamic damping is not included. The unit of this parameter depends on the Units parameter.

Dependencies

This parameter is visible when the Model parameter is set to Coupled flap and lag or Isolated lag.

Programmatic Use

Block Parameter: lagDamping

Type: double

Values: real nonnegative scalar or vector of length Nb

Default: 4067.5

Algorithms

This block computes rigid‑blade flap and lag dynamics using a small‑angle formulation. The block operates on distributed aerodynamic force inputs to compute flap and lag moments and integrates the corresponding equations of motion.

Rotor speed, Ω, is obtained either as a block parameter (when Rotor Rotational Speed Source is set to Dialog) or as an input signal (when set to Port).

Distributed force inputs are provided over the blade span as normal (f_z) and in‑plane (f_x) forces per unit span. The inputs are sized N_b-by-N_r, where N_b is the number of blades and N_r is the number of radial segments.

The blade span is discretized based on the Radial locations source parameter.

For a uniform distribution, the span is divided into N_r equal segments. For custom distribution, the input to the parameter ‘Non dimensional radial element edges’ is considered as the limits of the blade elements. For example, if the user specifies radial node locations {r₀, r₁,…,r_Nr}, the segment lengths will be computed as dr_i= r_i+1 -r_i.

It is assumed that the provided force values are at the mid points of these segments. The dimensional mid points of the segments will be y_i = R(r_i +dr_i/2)

For each blade b, flap and lag moments are computed by integrating contributions from all radial segments:

$$\begin{array}{l}{M}_{\beta }={\displaystyle {\int }_{e_{\beta }R}^R{f}_z(y-e_{\beta }R)}dy\\ M_{\varsigma }={\displaystyle {\int }_{e_{\varsigma }R}^R{f}_x(y-e_{\varsigma }R)}dy\end{array}$$

Note that only the force contributions from the portion of the blade beyond the flap hinge offset (e_β) is considered in computation of flap moment. Similarly, for lag moment computation, only the portion of the blade beyond the lag hinge offset (e_ζ) is considered. Also, in the block implementation, flap and lag moments are computed by summing the contributions from discrete blade segments rather than continuous integration.

The rigid‑blade equations of motion are second‑order ordinary differential equations defined per blade. For the Coupled Flap and Lag mode:

$$\begin{array}{l}{I}_{\beta }(\ddot{\beta }+v_{\beta }^2{\Omega }^2\beta -2\Omega \beta \dot{\zeta })+C_{\beta }\dot{\beta }=K_{\beta }{\beta }_p+{\displaystyle {\int }_{eR}^R{f}_z(y-e_{\beta }R)}dy\\ I_{\varsigma }(\ddot{\zeta }+v_{\zeta }^2{\Omega }^2\zeta +2\Omega \dot{\beta }\beta )+C_{\zeta }\dot{\zeta }={\displaystyle {\int }_{eR}^R{f}_x(y-e_{\zeta }R)}dy\end{array}$$

The state vector is defined as $$x={\left[\begin{array}{cccc}\beta & \dot{\beta }& \zeta & \dot{\zeta }\end{array}\right]}^{\text{'}}$$, with each component of size N_b​×1. The total number of states is 4N_b​ for the coupled model and 2_b​ for isolated flap or isolated lag models. The equations of motion are numerically integrated to obtain blade flap and/or lag angles and rates. Distributed force inputs are updated at each time step and used to recompute the corresponding moments.

For Isolated flap mode, only the first equation is solved, with the coupling term omitted. For Isolated lag mode, only the second equation is solved, with Coriolis terms omitted.N

References

[1] Leishman, Gordon J. Principles of Helicopter Aerodynamics with CD Extra. Cambridge University Press, 2006.

[2] Johnson, Wayne. Rotorcraft Aeromechanics. Vol. 36. Cambridge University Press, 2013.

Version History

Introduced in R2026b